Prove Cauchy Sequence Convergence: {p_n} \rightarrow p

In summary, if {p_n} is a Cauchy sequence and there is a subsquence {p_{n_i}} and a number p such that p_{n_i} \rightarrow p, then the full sequence converges to p as well. This is shown by taking \varepsilon > 0, choosing N such that n_k, n > N implies d(p_{n_k},p) < \frac{\varepsilon}{2} and d(p_n,p_{n_k}) < \frac{\varepsilon}{2}, and using the definition of convergence to prove that d(p_n,p) < \varepsilon.
  • #1
mynameisfunk
125
0

Homework Statement


Suppose that {[tex]p_n[/tex]} is a Cauchy sequence and that there is a subsquence {[tex]p_{n_i}[/tex]} and a number [tex]p[/tex] such that [tex]p_{n_i} \rightarrow p[/tex]. Show that the full sequence converges, too; that is [tex]p_n \rightarrow p[/tex].


Homework Equations





The Attempt at a Solution


Take [tex]\varepsilon > 0[/tex]. take [tex]N[/tex] s.t. [tex]n_k,n > N[/tex] implies that [tex]d(p_{n_k},p)< \frac{\varepsilon}{2}, d(p_n,p_{n_k}) < \frac{\varepsilon}{2}[/tex]. Hence [tex]d(p_n,p) \leq d(p_{n_k},p)+d(p_{n_k},p_n) \leq \varepsilon[/tex] Thus {[tex]p_n[/tex]} converges to [tex]p[/tex].
 
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  • #2
Actually, [tex]d(p_n,p) \leq d(p_{n_k},p)+d(p_{n_k},p_n) < \varepsilon[/tex].
 
  • #3
isnt that what i wrote?
 
  • #4
Is the only thing i need to fix the scrictly less than inequality??
 
  • #5
mynameisfunk said:
Is the only thing i need to fix the scrictly less than inequality??

Yes. Without strictly less, you're not consistent with the definition of convergence.
 

Related to Prove Cauchy Sequence Convergence: {p_n} \rightarrow p

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses. In other words, for any small positive number, there exists a point in the sequence after which all the terms are within that distance of each other.

2. What does it mean for a Cauchy sequence to converge?

A Cauchy sequence converges if the terms of the sequence approach a single limit as the sequence progresses. In other words, the terms become closer and closer together and eventually "converge" to a single value.

3. How do you prove that a Cauchy sequence is convergent?

To prove that a Cauchy sequence is convergent, we need to show that the terms of the sequence get arbitrarily close to each other, and that they also get arbitrarily close to a specific limit. This can be done using the definition of a Cauchy sequence and the limit of a sequence.

4. What is the significance of proving Cauchy sequence convergence?

Proving that a Cauchy sequence is convergent is important because it guarantees that the sequence has a well-defined limit, which can be useful in many mathematical and scientific applications. Additionally, it shows that the sequence is "complete" in the sense that it does not have any gaps or missing points.

5. Can a Cauchy sequence fail to converge?

Yes, a Cauchy sequence can fail to converge. This can happen if the sequence is not defined for all values, or if it does not get arbitrarily close to a specific limit. In other words, the terms of the sequence do not approach a single value as the sequence progresses.

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